"... Cutting planes are often used as an efficient means to tighten continuous relaxations of mixed-integer optimization problems and are a vital component of branch-and-cut algorithms. A critical step of any cutting plane algorithm is to find valid inequalities, or cuts, that improve the current relaxat ..."

Cutting planes are often used as an efficient means to tighten continuous relaxations of mixed-integer optimization problems and are a vital component of branch-and-cut algorithms. A critical step of any cutting plane algorithm is to find valid inequalities, or cuts, that improve the current relaxation of the integer-constrained problem. The maximally violated valid inequality problem aims to find the most violated inequality from a given family of cuts. $k$-projection polytope constraints are a family of cuts that are based on an inner description of the cut polytope of size $k $ and are applied to $k\times k $ principal minors of a semidefinite optimization relaxation. We propose a bilevel second order cone optimization approach to solve the maximally violated $k $ projection polytope constraint problem. We reformulate the bilevel problem as a single-level mixed binary second order cone optimization problem that can be solved using off-the-shelf conic optimization software. Additionally we present two methods for improving the computational performance of our approach, namely lexicographical ordering and a reformulation with fewer binary variables. All of the proposed formulations are exact. Preliminary results show the computational time and number of branch-and-bound nodes required to solve small